In this section, we give examples of simple settings where changes are easily detectable but where using cross-validation with squared error loss can lead to both under- and over-estimation of the number of change-points. For these negative results, we focus on the univariate mean change-point regression problem (REF ), where additionally \(\varepsilon _i \sim \mathcal {N}(0, \sigma ^2)\) . Furthermore, we consider a version of two-fold cross-validation, termed the \(\operatorname{COPPS}\)  procedure in [1]}, with optimal partitioning. However it will be clear that our constructions highlighting the undesirable properties of cross-validation, and our conclusions, can be generalised to other settings and other forms of cross-validation employing squared error loss.
